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point constraints in a first order traffic model
Edda Dal Santo, Carlotta Donadello, Sabrina F. Pellegrino, Massimiliano D.
Rosini
To cite this version:
Edda Dal Santo, Carlotta Donadello, Sabrina F. Pellegrino, Massimiliano D. Rosini. Representation of capacity drop at a road merge via point constraints in a first order traffic model. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2019, 53 (1), pp.1-34. �10.1051/m2an/2019002�.
�hal-02922756�
https://doi.org/10.1051/m2an/2019002 www.esaim-m2an.org
REPRESENTATION OF CAPACITY DROP AT A ROAD MERGE VIA POINT CONSTRAINTS IN A FIRST ORDER TRAFFIC MODEL
Edda Dal Santo
1, Carlotta Donadello
2,∗, Sabrina F. Pellegrino
3, and Massimiliano D. Rosini
4,5Abstract.We reproduce thecapacity drop phenomenonat a road merge by implementing a non-local point constraint at the junction in a first order traffic model. We call capacity drop the situation in which the outflow through the junction is lower than the receiving capacity of the outgoing road, as too many vehicles trying to access the junction from the incoming roads hinder each other. In this paper, we first construct an enhanced version of the locally constrained model introduced by Hautet al. (Proceedings 16th IFAC World Congress. Prague, Czech Republic229(2005) TuM01TP/3), then we propose its counterpart featuring a non-local constraint and finally we compare numerically the two models by constructing an adapted finite volumes scheme.
Mathematics Subject Classification. 35L65, 35R02, 90B20, 76M12.
Received March 12, 2018. Accepted December 28, 2018.
1. Introduction and main results
In macroscopic vehicular traffic modeling a particular attention is devoted to the dynamics of crossroads, as it is the essential building block to the modeling of traffic in a road network. From the mathematical point of view, the basic model for a crossroad is given by a system of conservation laws on an oriented star shaped graph. In this setting, the mere assumption of conservation of the number of vehicles through the junction is not enough to ensure the uniqueness of solutions. Further conditions, depending on the specific situation we aim at describing, should be imposed. This explains the huge number of such models available in the literature, see for instance [8,12,19–22] and the references therein.
The aim of this paper is to introduce and compare simple first order models able to reproduce the capacity drop phenomenon at a merge. We callcapacity drop the situation in which the outflow through the junction is
Keywords and phrases.Scalar conservation law, LWR model, traffic flow on networks, point constraint on the flux, finite volumes schemes.
1 Dipartimento di Ingegneria e Scienze de l’Informazione e Matematica, Universit`a dell’Aquila, Via Vetoio, 67100 L’Aquila, Italy.
2 Laboratoire de math´ematiques, CNRS UMR 6623, Universit´e de Bourgogne Franche-Comt´e, 16 route de Gray, 25030 Besan¸con, France.
3 Dipartimento di Matematica, Universit`a di Bari, Via E. Orabona 4, 70126 Bari, Italy.
4 Dipartimento di Matematica e Informatica, Universit`a di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy.
5 Uniwersytet Marii Curie-Sk lodowskiej, Plac Marii Curie-Sk lodowskiej 1, 20-031 Lublin, Poland.
∗Corresponding author:carlotta.donadello@univ-fcomte.fr
Article published by EDP Sciences c EDP Sciences, SMAI 2019
lower than the receiving capacity of the outgoing road, as too many vehicles trying to access the junction from the incoming roads hinder each other.
To the best of our knowledge, the first model aiming to capture such phenomenon is the Haut-Bastin-Chitour (HBC) model introduced in [22]. Roughly speaking, the authors introduce a decreasing function of the sending capacity of the incoming roads to bound the receiving capacity of the outgoing one. This model suffers two drawbacks: from the modeling point of view, it can produce a persistent traffic jam even starting from initial conditions which lead to moderate transient congestion in the unconstrained model; from the mathematical point of view, this model is not consistent, see Definition 2.6. In the present paper, we first construct an enhanced version of the HBC model, then we propose a new model in which the capacity drop is reproduced by imposing a non-local point constraint at the junction, and finally compare numerically the two models.
The non-local model can be seen as the natural generalization to the framework of a merge of the approach used in [2–5], where the authors developed analytical and numerical tools for the representation of capacity drop in vehicular and pedestrian traffic models on a single road by means of non-local point constraints on the flux. One of the main advantages of using non-local point constraints instead of local point constraints, as it was done in [9–11,13–17], is that the non-local approach allows for a more realistic representation of the transient behavior between congested traffic and free flow, see for instance [3] where the model presented in [16]
is compared with its non-local counterpart. We stress that the introduction of a non-local point constraint in the model does not substitute the implementation of a ramp metering strategy to avoid the formation of a traffic jam, as the description of the traffic evolution and its control are of course separate issues.
We construct a finite volumes scheme by adapting the finite volumes method introduced in [1] to the con- strained case, similarly to what has been done in [3]. We validate our scheme and implementation for both the local and non-local constraint case by comparison with explicit solutions here computed. Our simulations show that, at least in these cases, the method converges numerically and that the implementation of a non-local point constraint allows for a more regular transition between congested and free traffic situations. Moreover, a qualitative analysis of the behavior of numerical solutions suggests a simple way to calibrate the parameters appearing in our non-local operator.
The paper is organized as follows. The next section is an introduction to the basic definitions and notations used in the paper. In Section3we outline the main features of the local model. In Section4we briefly comment on the properties of the non-local model. Section5 is devoted to the description and validation of the adapted finite volumes numerical scheme. Section 5.3 presents a numerical comparison of local and non-local models.
Section6contains lengthy computations and technical proofs. The last section is devoted to concluding remarks.
2. Basic definitions and notations
In this paper we consider a junction (crossroad) connecting two incoming and one outgoing roads. In terms of graph theory, we consider a semi-infinite star-graph with two incoming and one outgoing edges.
The incoming roads are parameterized by x ∈ (−∞,0] and numbered by the index i ∈ I = {1,2}, while the outgoing road is parameterized by x ∈ [0,∞) and numbered by the index 3. In both parameterizations the junction is located at x = 0. We denote the generic road by Ωh, h∈ H= {1,2,3}, and the network by N = Πh∈HΩh.
On each road the traffic evolution is described by the Lighthill-Whitham-Richards (LWR) model [24,27], namely by a scalar conservation law of the form
∂tρh+∂xfh(ρh) = 0, t >0, x∈Ωh, (2.1)
where ρh is the density of vehicles and fh is the flux along the hth road. We augment (2.1) with the initial condition
ρh(0, x) =ρh,0(x), x∈Ωh, (2.2)
where ρh,0 is assumed to be in L1∩BV(Ωh; [0, ρmax]). We assume that the roads have a common maximal densityρmax>0 and the fluxesfhare bell-shaped (unimodal). More precisely we assume that
fh belongs to Lip([0, ρmax]; [0, fhmax]), fh(0) = 0 = fh(ρmax) and there
existsρh,c∈(0, ρmax) such thatfh0(ρ) (ρh,c−ρ)>0 fora.e.ρ∈[0, ρmax]. (F) Above fhmax is the maximum value of fh in [0, ρmax] and we have fhmax = fh(ρh,c). Assumption (F) is compatible with the standard LWR traffic model, see [8,19,20,28], and appears in most papers devoted to conservation laws subject to point constraints [1,4,6,9,13] as it ensures the existence of strong traces of solutions on both sides of the constraint location, see Theorem 2.2 of [6] which is a reformulation of the results of [25,29]. The same hypothesis has also been used in [22], which is the starting point of our analysis.
Let ¯ρhbe the trace ofρh atx= 0,h∈H, andγ(~ρ) = ( ¯ρ1,ρ¯2,ρ¯3) the vector of the traces of the densities at the junction. We use the following definition of weak solution on the network.
Definition 2.1. The vector function ~ρ = (ρ1, ρ2, ρ3), where ρh : (0,∞)×Ωh → [0, ρmax], h ∈ H, is a weak solution to (2.1) and (2.2),h∈H, in the networkN if
– ρh∈C0((0,∞); L1(Ωh; [0, ρmax]))∩BVloc((0,∞)×Ωh; [0, ρmax]),h∈H;
– for i ∈ I, ρi is a weak entropy solution to (2.1) and (2.2)h=i, namely for every c ∈ [0, ρmax] and every nonnegative test functionφ∈C∞(R×(−∞,0);R) with compact support
Z ∞ 0
Z
Ωi
|ρi−c|∂tφ+ sign(ρi−c) fi(ρi)−fi(c)
∂xφ
dxdt+ Z
Ωi
|ρi,0(x)−c|φ(0, x) dx≥0;
– ρ3 is a weak entropy solution to (2.1) and (2.2)h=3, namely for everyc ∈ [0, ρmax] and every nonnegative test functionφ∈C∞(R×(0,∞);R) with compact support
Z ∞ 0
Z
Ω3
|ρ3−c|∂tφ+ sign(ρ3−c) f3(ρ3)−f3(c)
∂xφ
dxdt+ Z
Ω3
|ρ3,0(x)−c|φ(0, x) dx≥0;
– the number of vehicles across the junction is conserved, namely
f3( ¯ρ3(t)) =f1( ¯ρ1(t)) +f2( ¯ρ2(t)), fora.e.t >0.
Since the above definition does not ensure uniqueness of weak solutions, we need some additional selection criteria in order to hope for well-posedness of the Cauchy problem. This is achieved by declaring which Riemann solver we adopt at the junction, see [8,12,19–22]. We recall the following definition of Riemann solver at the junction.
Definition 2.2. We say that
RS= (RS1,RS2,RS3) : Λ→BV(N; Λ), Λ = [0, ρmax]3,
is a Riemann solver at the junction if for any constant initial datumρ~0= (ρ1,0, ρ2,0, ρ3,0)∈Λ the map (t, ~x)7→ RS[~ρ0](~x/t) = (RS1[~ρ0](x1/t),RS2[~ρ0](x2/t),RS3[~ρ0](x3/t))
is a self-similar weak solution to (2.1) and (2.2),h∈H, in the networkN.
In Definition2.6below we state some natural properties for Riemann solvers at the junction. In order to do so, we have to recall some notation from the literature on traffic modeling.
Definition 2.3(Def. 2.6 in [8]). We distinguish betweengood andbad (initial)data as follows:
max max
max 3
3 3
max
Figure 1.The equilibrium demand and supply functions.
– fori∈I,ρi,0∈[0, ρmax] is a good datum ifρi,0≥ρi,c; – ρ3,0∈[0, ρmax] is a good datum if ρ3,0≤ρ3,c;
– forh∈H,ρh,0∈[0, ρmax] is a bad datum if it is not a good datum.
With a slight abuse of notation, in the following we writeρh,0∈Gif it is a good datum,~ρ0= (ρ1,0, ρ2,0, ρ3,0)∈ GGGif each component of~ρ0 is a good datum, and so on.
Definition 2.4 (Sect. 5.2.3 in [20]). Fori∈I, theequilibrium demand function (sometimes calledequilibrium sending capacity) of theith incoming road Ωi is the map ∆i: [0, ρmax]→[0, fimax], see Figure1, defined by
∆i(ρ) =
(fimax ifρis a good datum, fi(ρ) otherwise.
Theequilibrium supply function (sometimes calledequilibrium receiving capacity) of the outgoing road Ω3 is the mapΣ3: [0, ρmax]→[0, f3max], see Figure1, defined by
Σ3(ρ) =
(f3max ifρ is a good datum,
f3(ρ) otherwise. (2.3)
Definition 2.5(Def. 2.5 in [8]). We say thatρ~0∈Λ is anequilibriumfor a Riemann solver at the junctionRS ifRS[~ρ0]≡~ρ0; as a consequence the constant valued function (t, x)7→~ρ0is a stationary solution.
Definition 2.6. LetRS: Λ→BV(N; Λ) be a Riemann solver at the junction.
– We say that RS has the property (P1) (see [21], Def. 8) if γ(RS[~ρ0]) = γ(RS[~ρ0∗]) for any initial data
~
ρ0, ~ρ0∗∈Λ such thatρh,0=ρh,0∗ wheneverρh,0 orρh,0∗ is a bad datum,h∈H.
– We say that RS is consistent (see [20], p. 72) if for any initial datum ~ρ0 ∈ Λ the vector of the traces γ(RS[~ρ0]) is an equilibrium forRS in the sense of Definition2.5.
– We say thatRS is L1loc-continuous at~ρ0∈Λ ifRShis L1loc-continuous atρh,0 for allh∈H.
In Definition2.7 below we introduce the general form of the Riemann solvers at the junction considered in this paper, namely we introduce the concept of admissibility. We use the following notation:
– forh∈H,Rhis the Lax Riemann solver [7] associated to (2.1);
– fori∈I, ˆρi∈C0([0, fimax]; [ρi,c, ρmax]) is the inverse function offi|[ρi,c,ρmax]; – ˇρ3∈C0([0, f3max]; [0, ρ3,c]) is the inverse function off3|[0,ρ3,c].
Figure 2.The geometrical meaning of (2.5) forα∈ {α0, α00, α000} ⊂[0,1] such thatα00, Q(~ρ0)<
Q(~ρ0)−∆2(ρ2,0)< α00Q(~ρ0)<∆1(ρ1,0)< α0Q(~ρ0). The gray area corresponds to the attainable values for (Γ1,Γ2), namely to the region {(Γ1,Γ2) ∈ [0,∆1(ρ1,0)]×[0,∆2(ρ2,0)] : Γ1+ Γ2 ≤ Q(~ρ0)}.The dotA0 represents the passing flow corresponding to the priority coefficientα0, and so on. In the figure we let ∆i= ∆i(ρi,0),i∈I.
Definition 2.7. For any fixed priority factor α ∈ [0,1] and receiving capacity Q : Λ → [0, f3max] such that Q(~ρ)≤Σ3(ρ3) for anyρ~∈Λ, we say that a Riemann solver at the junction Rj : Λ→BV(N; Λ) isadmissible if it has the form
~
ρ07→(R1[ρ1,0,ρˆ1(Γ1(~ρ0))],R2[ρ2,0,ρˆ2(Γ2(~ρ0))],R3[ ˇρ3(Γ1(~ρ0) + Γ2(~ρ0)), ρ3,0]), (2.4) where Γi : Λ → [0, fimax], i ∈ I, are the passing flow at the junction from theith road corresponding to the receiving capacityQ=Q(~ρ), and are defined by
Γ1≡
∆1 if ∆1+ ∆2≤Q,
∆1 ifαQ≥∆1,
αQ ifQ−∆2< αQ <∆1, Q−∆2 ifαQ≤Q−∆2,
otherwise, Γ2≡
(∆2 if ∆1+ ∆2≤Q,
Q−Γ1 otherwise, (2.5)
where ∆i stands for ∆i(ρi,0),i∈I.
In other words, for~ρ0∈Λ, we have that Γi(~ρ0),i∈I, are defined as follows:
– if the total sending capacity of the incoming roads ∆1(ρ1,0) + ∆2(ρ2,0) does not exceedQ(~ρ0), then Γi(~ρ0) =
∆i(ρi,0),i∈I;
– otherwise, the passing flow at the junction Γ1(~ρ0) + Γ2(~ρ0) coincides with Q(~ρ0) and is split between the incoming roads in accordance with the priority factor α, see Figure 2.
We observe that Γ1defined in (2.5) can be rewritten in a more compact form as follows Γ1≡
(∆1 if ∆1+ ∆2≤Q, max{Q−∆2,min{αQ,∆1}} if ∆1+ ∆2> Q.
Clearly, in the present setting choosing an admissible Riemann solver at the junction is equivalent to choosing a priority factorα∈[0,1] and a receiving capacityQ: Λ→[0, f3max] such thatQ(~ρ)≤Σ3(ρ3) for anyρ~∈Λ.
Notice that an admissible Riemann solver at the junction Rj associates to any road-wise constant initial condition~ρ0∈Λ, the self-similar weak solutionRj[~ρ0] in the networkN realizing the maximum of the passing flow at the junction because
Γ1(~ρ0) + Γ2(~ρ0) =
(∆1(ρ1,0) + ∆2(ρ2,0) if ∆1(ρ1,0) + ∆2(ρ2,0)≤Q(~ρ0),
Q(~ρ0), otherwise. (2.6)
Remark 2.8. The output of an admissible Riemann solver at the junctionRj can be understood as a “collec- tion” of solutions to three initial-boundary value problems (one for each road) coupled through their boundary conditions
∂tρi+∂xfi(ρi) = 0, t >0, x∈Ωi, ρi(0, x) =ρi,0, x∈Ωi, ρi(t,0) = ˆρi(Γi), t >0,
i∈I,
∂tρ3+∂xf3(ρ3) = 0, t >0, x∈Ω3, ρ3(0, x) =ρ3,0, x∈Ω3, ρ3(t,0) = ˇρ3(Γ1+ Γ2), t >0.
(2.7)
We recall that the solutions to the initial-boundary value problems (2.7) coincide with the restrictions to Ωh
of the Kruzhkov [23] entropy solutions, constructed via the Lax Riemann solver, to the Riemann problems
∂tρi+∂xfi(ρi) = 0, t >0, x∈R, ρi(0, x) =
(ρi,0 ifx <0, ˆ
ρi(Γi) ifx≥0,
i∈I,
∂tρ3+∂xf3(ρ3) = 0, t >0, x∈R, ρ3(0, x) =
(ρ3,0 ifx <0, ˇ
ρ3(Γ1+ Γ2) ifx≥0,
(2.8) respectively. We observe that, by (2.5) and (2.6), for any~ρ0∈Λ we have
Γi(~ρ0)≤∆i(ρi,0), i∈I, Γ1(~ρ0) + Γ2(~ρ0)≤Q(~ρ0)≤Σ3(ρ3,0).
Therefore the tracesγ(Rj[~ρ0]) = ( ¯ρ1,ρ¯2,ρ¯3) satisfy fora.e.t >0
fi( ¯ρi(t)) = Γi(~ρ0), i∈I, f3( ¯ρ3(t)) = Γ1(~ρ0) + Γ2(~ρ0), (2.9) but not necessarily ¯ρi(t) = ˆρi(Γi),i∈I, or ¯ρ3(t) = ˇρ3(Γ1+ Γ2), see [7].
With a slight abuse of notations, we denote by Rh, h ∈ H, the Lax Riemann solvers associated to the initial-boundary value problems (2.7) or to the Riemann problems (2.8).
In Definition 2.9 below we introduce three admissible Riemann solvers at the junction. Each of them is characterized by a different receiving capacityQ. From now on, we assume that
f3max< f1max+f2max, (2.10)
and we introduce theconstraint function g: [0, f1max+f2max]→[0, f3max] defined by
g(s) =
f3max ifs≤f3max,
f3max+gminb−f−fmax3max 3
(s−f3max) iff3max< s < b,
gmin otherwise,
(2.11)
whereb∈(f3max, f1max+f2max] andgmin∈(0, f3max).
Definition 2.9. Letα∈[0,1] be a priority factor and gbe a constraint function as in (2.11).
– We denote byRCGPj : Λ → BV(N; Λ) the admissible Riemann solver at the junction introduced in [12]
and corresponding to the receiving capacityQ≡Σ3, whereΣ3is the equilibrium receiving capacity defined by (2.3).
– We denote byRHBCj : Λ→BV(N; Λ) the admissible Riemann solver at the junction introduced in [22] and corresponding to the effective receiving capacity QHBC: Λ→[0, f3max] defined by
QHBC(~ρ0) = min{Σ3(ρ3,0), g(∆1(ρ1,0) + ∆2(ρ2,0))}. (2.12)
– We denote byRlj: Λ→BV(N; Λ) the admissible Riemann solver at the junction corresponding to thelocal effective receiving capacity Ql: Λ→[0, f3max] defined by
Ql(~ρ0) = min
QHBC(~ρ0), QHBC(T[~ρ0]), QHBC(T2[~ρ0]) , (2.13) where T =γ◦ RHBCj .
We observe that by definition Ql(~ρ0)≤QHBC(~ρ0)≤Σ3(ρ3,0)≤f3max. In particular, this ensures that the Riemann solvers at the junction introduced in Definition2.9are admissible.
Remark 2.10. We stress that, whenever ∆1(ρ1,0) + ∆2(ρ2,0) is smaller than f3max, the effective receiving capacityQHBC(~ρ0) coincides with the equilibrium receiving capacityΣ3(ρ3,0) andRHBCj [~ρ0]≡ RCGPj [~ρ0].
The solverRCGPj does not represent any capacity drop effect and for this reason Haut, Bastin and Chitour introducedRHBCj , see [22]. Roughly speaking, the Riemann solver at the junctionRHBCj accounts for the capacity drop effect by takingQ≡QHBC(~ρ0) instead of Q≡Σ3(ρ3,0) in Definition2.7.
We list below two drawbacks ofRHBCj .
(D.I) The main drawback (at least from the mathematical point of view) is that RHBCj is not consistent, see Section 6.1 for an explicit example. Roughly speaking, let γ(~ρ) = ( ¯ρ1,ρ¯2,ρ¯3) be the vector of traces at x= 0 of the solution ~ρ=RHBCj [~ρ0] corresponding to an initial datum ~ρ0 ∈BBG such thatΣ3(ρ3,0)≤
∆1(ρ1,0) + ∆2(ρ2,0)< b. By definition QHBC(~ρ0) =g(∆1(ρ1,0) + ∆2(ρ2,0))≤f3max =Σ3(ρ3,0). It may happen that γ(~ρ) is not an equilibrium. If, fori ∈ I, ¯ρi = ˆρi, which by definition is good datum, then g(∆1( ¯ρ1) + ∆2( ¯ρ2))< g(∆1(ρ1,0) + ∆2(ρ2,0)) because g is decreasing. Thus the constraint diminishes andγ(~ρ) does not satisfy it.
(D.II) As already observed in [22], the solution associated to RHBCj may develop a traffic jam that persists forever, even if the same initial condition leads to very moderate congestion in the solution associated to RCGPj (without capacity drop representation).
We fix the drawback (D.I) by introducingRlj, see Section3.
In the setting of crowd dynamics, the model proposed in [16], featuring a point constraint depending on a point value of the density, suffers from a problem similar to (D.II). In [5] the authors showed that this issue can be overcome by considering a point constraint whose value at each timet >0 depends on the average value of the solution on an interval. In this paper we propose an analogous approach to obtain a more realistic representation of the transient behavior between congested and free traffic at a merge. Roughly speaking, according to the model we describe in Section4the effective receiving capacity of the junction depends on the average density of vehicles on the incoming roads in an upstream neighborhood of the intersection, and not merely on the traces of the density functions atx= 0.
In the following theorem we collect the main properties of the admissible Riemann solvers at the junction introduced in Definition2.9; the proof is deferred to Section6.
Theorem 2.11. – The Riemann solver at the junction RCGPj has the property (P1), is consistent, is L1loc- continuous, but does not reproduce the capacity drop at the junction.
– The Riemann solver at the junctionRHBCj has the property (P1), reproduces the capacity drop at the junction but is not consistent.
– The Riemann solver at the junctionRlj has the property (P1), is consistent and reproduces the capacity drop at the junction but it fails to beL1loc-continuous.
Remark 2.12. In the proof of Theorem2.11we give an explicit example to show thatRljis not L1loc-continuous.
We can observe that in the same situation RHBCj is L1loc-continuous, but we do not address the L1loc-continuity of the solverRHBCj in this paper.
3. The admissible Riemann solver at the junction R
ljIn this section we give a more explicit description ofRlj. As already pointed out in (D.I), we introduce Rlj to overcome the non-consistency of the Riemann solverRHBCj . The lack of such property has consequences not only from a mathematical point of view (as it means that the solver provides non-stable solutions) but also in numerical simulations. Indeed, if we implementRHBCj in a finite volumes numerical scheme we do not observe the expected solution, as it is destroyed after a few time iterations, but we observe the solution corresponding toRlj, see Section5for the description of our scheme. Therefore, it makes sense to introduce the iterate version ofRHBCj and study its properties.
We recall that our analysis, which relies on a case-by-case study, heavily depends on assumptions (2.10) and (2.11). In particular, we prove that, forRlj, Definition2.9is equivalent to say thatRlj=RHBCj ◦ T2, where T is the composition of the Riemann solverRHBCj and the trace operatorγ, that is
T =γ◦ RHBCj : Λ → Λ,
~
ρ07→γ(RHBCj [~ρ0]). (3.1)
Moreover, by the same analysis we have the consistency of Rlj and a more explicit definition ofRlj, which associates to any initial condition the corresponding solution without a direct computation of the iterations of T, see Proposition3.1.
If assumptions (2.10) and (2.11) are not enforced, three iterations might not be sufficient to achieve consis- tency, additional cases need to be discussed and, of course, Proposition3.1does not hold.
Whenever the following quantities make sense, we denote ˆ
ρ1,g= ˆρ1(gmin−f2max), ρˆ2,g= ˆρ2(gmin−f1max), ρˇ3,g= ˇρ3(gmin), ˆ
ρ1,α= ˆρ1(α gmin), ρˆ2,α= ˆρ2((1−α)gmin),
~
ρA= ( ˆρ1,g, ρ2,c,ρˇ3,g), ~ρB= ( ˆρ1,α, ρˆ2,α,ρˇ3,g), ~ρC= (ρ1,c,ρˆ2,g,ρˇ3,g).
Notice that by (2.10) and (2.11) we have gmin ≤ f3max < f1max+f2max, whence gmin−f2max < f1max and gmin−f1max< f2max.
Proposition 3.1. The Riemann solver Rlj : Λ→BV(N; Λ)behaves as follows.
(a) If (ρ1,0, ρ2,0)∈BG,QHBC(~ρ0) =g(f1(ρ1,0) +f2max)andf1(ρ1,0)< α QHBC(~ρ0), then Ql(~ρ0) =QHBC(~ρ0),
and
γ◦ Rlj[~ρ0] =T[~ρ0] = ρ1,0,ρˆ2 Ql(~ρ0)−f1(ρ1,0)
,ρˇ3 Ql(~ρ0) . (b) If (ρ1,0, ρ2,0)∈GB,QHBC(~ρ0) =g(f1max+f2(ρ2,0))andf2(ρ2,0)<(1−α)QHBC(~ρ0), then
Ql(~ρ0) =QHBC(~ρ0), and
γ◦ Rlj[~ρ0] =T[~ρ0] = ˆρ1 Ql(~ρ0)−f2(ρ2,0)
, ρ2,0,ρˇ3 Ql(~ρ0) .
(c) If (ρ1,0, ρ2,0)∈BB,QHBC(~ρ0)> g(f1(ρ1,0) +f2max)andf1(ρ1,0)< αg(f1(ρ1,0) +f2max), then Ql(~ρ0) =QHBC(T[~ρ0]) =g(f1(ρ1,0) +f2max),
and
γ◦ Rlj[~ρ0] =T[~ρ0] = ρ1,0,ρˆ2 Ql(~ρ0)−f1(ρ1,0)
,ρˇ3 Ql(~ρ0) .
Table 1. The correspondence between the cases of Proposition3.1and those of Proposition6.2.
Proposition3.1 Proposition6.2 Case (a) (BGG.i-1) (BGB.iii-1) Case (b) (GBG.i-3) (GBB.iii-3)
Case (c) (BBG.ii-1) (BBB.iii-1) (BBB.iv-1) Case (d) (BBG.ii-4.a) (BBB.iii-4.a) (BBB.iv-6.a)
Case (e) (GGB.i) (BGB.iv-1) (GBG.iv-2) (BGB.iv-4) (BBB.iv-3) (BBB.iv-4) (BGG.ii) (BBG.i) (BGB.i) (BGB.ii) (BBB.ii) (BBB.i) (GBG.ii) (GBB.i) (GBB.ii) (GBB.iv-1) (GBB.iv-2) (GBB.iv-4) (BBB.iv-6.b)
Case (f) (GGG) (BGG.i-2) (BGG.i-3) (BGG.i-4) (GGB.ii) (BBG.ii-2) (BBG.ii-3) (BGB.iii-2) (BGB.iii-3) (BGB.iv-3) (BGB.iv-5) (BBB.iii-2) (BBB.iii-3) (BBB.iv-2) (BBB.iv-5) (BBG.ii-4.b) (BBB.iii-4.b) (BBB.iv-6.c) (GBG.i-2) (GBG.i-1) (GBG.i-4) (GBB.iii-1) (GBB.iii-2) (GBB.iv-3) (GBB.iv-5)
(d) If (ρ1,0, ρ2,0)∈BB,QHBC(~ρ0)> g(f1max+f2(ρ2,0))andf2(ρ2,0)<(1−α)g(f1max+f2(ρ2,0)), then Ql(~ρ0) =QHBC(T[~ρ0]) =g(f1max+f2(ρ2,0)),
and
γ◦ Rlj[~ρ0] =T[~ρ0] = ˆρ1 Ql(~ρ0)−f2(ρ2,0)
, ρ2,0,ρˇ3 Ql(~ρ0) . (e) If Σ3(ρ3,0)≤min{∆1(ρ1,0) + ∆2(ρ2,0), g(∆1(ρ1,0) + ∆2(ρ2,0))}, then
Ql(~ρ0) =QHBC(~ρ0) =Σ3(ρ3,0), andRlj[~ρ0]≡ RHBCj [~ρ0]≡ RCGPj [~ρ0].
(f) In all other cases Ql(~ρ0) =gmin and
γ◦ Rlj[~ρ0] =T[~ρ0] =
~
ρA if α gmin∈[0, gmin−f2max],
~
ρB if α gmin∈(gmin−f2max, f1max),
~
ρC if α gmin∈[f1max, f3max].
The proof consists of the case study deferred to Section6.3. For the reader’s convenience we summarize in Table1 the correspondence between the case studies and the points listed in Proposition3.1.
3.1. An explicit admissible solution for the local model
This section is devoted to the computation of an explicit solution by means of the local solver Rlj. We use such solution in Section5.2.1to perform a convergence analysis of our finite volumes numerical scheme subject to a local point constraint.
We consider f(ρ) ≡fh(ρ) = ρ(1−ρ) as the flux for each road. As initial condition, we choose ρ1,0(x) = χ[−1/2,0](x), ρ2,0(x) = 3/4χ[−1/4,0](x) and ρ3,0(x) = 0. We fix the priority factorα= 1/2 and the constraint function
g(s) =
(1/4 ifs≤1/4,
3−4s
8 if 1/4≤s≤1/2.
The exact solution in Figure3is obtained by an explicit analysis of the wave-front interactions, with computer assisted computation of front slopes and interaction times. Everywhere in the following we denote byσ(uL, uR) the speed of a shock connecting the left state uL to the right state uR, computed according to the Rankine- Hugoniot condition.
(a)The solution on Ω1. (b)The solution on Ω2. (c)The solution on Ω3
Figure 3. The solution in the (x, t)-plane obtained in Section3.1.
At t = 0, the local effective receiving capacity Ql is equal to gmin = g(1/2) = 1/8. Therefore, on Ω1 a rarefactionRO,1starts fromO(0,0) and its values are given by
RO,1(t, x) =1 2
1−x
t
, for −t≤x <−
√3 2 t.
On Ω2starts the backward shockSO,2 given by SO,2: ˙x(t) =σ 1
2 1 +
√3 2
! ,3/4
!
, x(0) = 0.
On Ω3a rarefaction starts fromO(0,0) and its values are given by RO,3(t, x) =1
2
1−x t
, for
√2
2 t < x≤t.
On Ω2, letB(xB, tB) be the point where the shock x(t) =−14+4t originated from (−1/4,0) interacts with the shockSO,2. As a result, fromB starts a shock given by
SB,2: ˙x(t) =σ 0, 1 2 1 +
√3 2
!!
, x(tB) =xB,
which reaches the junction inx= 0 at timet=tC= 3 that corresponds to the time at which the second incoming road becomes empty. On Ω1, inA(−1/2,1/2), the stationary shock originated from (−1/2,0) interacts with the rarefactionRO,1. As a result, fromAstarts a shock SA,1given by
SA,1: ˙x(t) =σ(0,RO,1(t, x(t))), x(1/2) =−1/2.
Let D(xD, tD) be the intersection between SA,1 and x(t) = −(√
3/2)t. From this point starts a forward shock
SD,1: ˙x(t) =σ 0, 1 2 1 +
√3 2
!!
, x(tD) =xD.
Att=tCthe local effective receiving capacityQlisg(1/4) = 1/4 and a rarefaction appears on Ω1. It is given by
RC,1(t, x) = 1 2
1− x t−tC
, for −
√ 3
2 (t−tC)< x≤0.
Moreover, on Ω3 at the same time starts a rarefactionRC,3given by RC,3(t, x) = 1
2
1− x t−tC
, for 0≤x <
√2
2 (t−tC).
LetF be the point whereSD,1 and RC,1 meet together. From this point starts a forward shock SF,1, with left state ρ= 0, which reaches the junction at evacuation timetG ≈4.25, then Ω1 is empty. Finally, on Ω3 at timetG starts a shock that interacts with the rarefactionRC,3 generating the shock
SG,3: ˙x(t) =σ(0, RC,3(t, x(t))), x(tG) = 0.
4. A junction model with non-local effective receiving capacity.
The main difference between the model in this section and the ones presented above lies in the algorithm used to compute the effective receiving capacity. For any given~ρ= (ρ1, ρ2, ρ3), withρh∈L1loc(Ωh; [0, ρmax]) we define thenon-local effective receiving capacity Qnl=Qnl(~ρ) by
Qnl(~ρ) = min{Σ3( ¯ρ3), g(∆1(ζ1) + ∆2(ζ2))}, (4.1) whereζi is a weighted average of the density of vehicles on Ωi in a neighborhood of the junction, namely
ζi= Z 0
−∞
wi(x)ρi(x) dx,
wherewi∈L∞(R−;R+) is an increasing function with compact support in [−`i,0] andkwikL1(R−)= 1,i∈I.
The concept of admissible solution introduced in the previous sections extends naturally in the following form.
Definition 4.1. Letα∈[0,1] be a priority factor and~ρ= (ρ1, ρ2, ρ3), withρh∈C0((0,∞); L1(Ωh; [0, ρmax]))∩ BVloc((0,∞)×Ωh; [0, ρmax]),h∈H, be a weak solution to (2.1) and (2.2) in the sense of Definition2.1. We say that~ρis an admissible solution of the non-local model if the following conditions, involving the vector of traces γ(~ρ) = ( ¯ρ1,ρ¯2,ρ¯3), hold fora.e.t:
f1( ¯ρ1(t)) =
∆1( ¯ρ1(t)) if ∆1( ¯ρ1(t)) + ∆2( ¯ρ2(t))≤Qnl,
∆1( ¯ρ1(t)) if αQnl≥∆1( ¯ρ1(t)),
αQnl if Qnl−∆2( ¯ρ2(t))< αQnl<∆1( ¯ρ1(t)), Qnl−∆2( ¯ρ2(t)) if αQnl≤Qnl−∆2( ¯ρ2(t)),
otherwise,
f2( ¯ρ2(t)) =
(∆2( ¯ρ2(t)) if ∆1( ¯ρ1(t)) + ∆2( ¯ρ2(t))≤Qnl, Qnl−f1( ¯ρ1(t)), otherwise,
where Qnl =Qnl(~ρ(t)) is thenon-local effective receiving capacity, computed on the profile of the solution at timetand defined by (4.1).
The analytical proof of existence and stability of such admissible solutions for a general Cauchy problem is a difficult open question. In this paper we limit our attention to special situations in which the initial condition is road-wise constant or the constraint functiongis piecewise constant.
Conjecture 4.2. Given a constraint function g as in (2.11) and fluxes fh, h ∈ H, satisfying (F), we can associate a unique admissible solution in C0((0,∞); Π3h=1L1(Ωh; [0, ρmax]))∩Π3h=1BVloc((0,∞)×Ωh; [0, ρmax]) to any road-wise constant initial condition in Λ. We denote by Sjnl : Λ→ C0([0, T]; Π3h=1L1(Ωh; [0, ρmax]))∩ Π3h=1BVloc((0,∞)×Ωh; [0, ρmax]) the solver operator.
The proof of this conjecture will appear in a separate paper together with an existence result for the Cauchy problem at a merge subject to piecewise constant time dependent point constraint at the junction.
The operator Sjnl is not a Riemann solver as in general it does not produce self-similar solutions. In fact the effective receiving capacity might change even if no new wave hits the intersection, just because the value of the average functions ζi is not constant in time. In the next example we see that, if the initial conditions are road-wise constant and the constraint function g is continuous decreasing as in (2.11), then the effective receiving capacity is a continuous function of time.
Example 4.3. Consider an initial datum~ρ0∈BBGsuch thatQnl(~ρ0) =g(∆1(ζ1(0))+∆2(ζ2(0))) =g(f1(ρ1,0)+
f2(ρ2,0)). If ¯ρi= ˆρifor at least one indexi∈I, the resulting waves are shocks with negative speedσi. In particular this means that ˆρi> ρ0,iand the average ζi(t) is strictly increasing in time.
As the speed of propagation of any wave in the solution is finite, there existsδ >0 such that if t < δ, then ζi(t) are still bad data (therefore ∆i(ζi(t)) =fi(ζi(t))) and we have
|ζi(t)−ζi(0)|= Z 0
−`i
wi(x) ˆρiχ[σit,0](x) +ρi,0χ(−∞,σit](x)−ρi,0 dx
= Z 0
−`i
wi(x)( ˆρi−ρi,0)χ[σit,0](x) dx≤ |fi(ρi,0)−fi( ˆρi)|kwkL∞(R−)t.
Hence, for any fixed ε > 0 we get |g(f1(ζ1(t)) +f2(ζ2(t)))−g(f1(ζ1(0)) +f2(ζ2(0)))| < ε as soon as t ≤ inf
δ, ε/ fimaxkwkL∞(R−) .
Remark 4.4. Property (P1) basically states that equilibria are determined by bad data, as substituting a good initial datum with a different good datum does not change the trace of solution. ForSjnl we can state an analogous property
For t >0 large enough Sjnl[~ρ0](t,0) =Sjnl[~ρ0∗](t,0) for any initial data ~ρ0, ~ρ0∗ ∈ Λ such
that ρh,0=ρh,0∗ wheneverρh,0 orρh,0∗ is a bad datum,h∈H. (P1nl) This property holds because the sending capacity of an incoming road which has an initial condition in G will stay constant forever (with valuefimax), no matter what happens on the other roads. This means that the effective receiving capacity will only depend on the sending and receiving capacities of the other roads.
We observe that the solution produced bySjnl on the incoming roads can only contain waves with negative speed. On the incoming roads, any wave of negative speed which has a good datum on the left needs to have on the right another good datum. The average of two good data is in their convex hull, so it is again a good datum. Therefore ifρ0,i∈Gthenζi(t) will also be inG, for allt≥0.
The fact that two different initial conditions for the outgoing road inGlead to the same asymptotic solution is straightforward.
If the functiong is piecewise constant, then the solution will be self-similar for a (possibly short) time span.
In this special caseSjnlcan be seen as a Riemann solver locally in time, see [2]. The case in whichgis piecewise constant is important in view of future investigation of the Cauchy problem for this model (a combination of the operator splitting method and the wave-front tracking algorithm leads to the construction of global solutions), and it is the only case in which a solution can be computed explicitely, as we do in the next section. The explicit solution we obtain is used to validate a finite volumes numerical scheme and investigate its numerical convergence in Section5.
4.1. An explicit admissible solution for the non-local model
In this section we compute an explicit solution by means ofSjnl in order to point out its properties.
(a)The solution on Ω1. (b)The solution on Ω2. (c)The solution on Ω3
Figure 4.The explicit solution in the (x, t)-plane obtained in Section4.1.
We consider f(ρ) ≡ fh(ρ) = ρ(1−ρ) as the flux for each road. As initial condition, we choose ρ1,0(x) = χ[−1/2,0](x),ρ2,0(x) = 3/4χ[−1/4,0](x) andρ3,0(x) = 0. We fix the priority factorα= 1/2, the constraint function
g(s) =
1/4 ifs≤ 14,
3
20 if 14 < s <209,
1
8 if 209 ≤s≤ 12, and the weight functionw(x) = 8(4x+ 1)χ[−1/4,0](x).
The exact solution is obtained by an explicit analysis of the wave-front interactions, with computer assisted computation of front slopes and interaction times presented in Figure4.
Notice that for time smaller than the time in which the non-local effective receiving capacity Qnl becomes 3/20, the explicit solution coincides with the solution computed in Section 3.1, therefore we omit it. With a slight abuse of notation we denote byt=tC ≈2.40 such time, which is obtained by solving the equation
∆1(ζ1(t)) + ∆2(ζ2(t)) = 9
20, t > tD= 8 (√
3 + 2)2·
At this time a further rarefaction appears in each of the incoming roads. Its values are given by RC,i(t, x) = 1
2
1− x t−tC
, for −
√3
2 (t−tC)< x≤ −
√70
10 (t−tC), i∈I.
Moreover, on Ω3 at the same time starts a rarefactionRC,3given by RC,3(t, x) = 1
2
1− x t−tC
, for
√10
5 (t−tC)< x≤
√2
2 (t−tC).
On Ω1, letEbe the point whereSD,1andRC,1meet together. From this point starts a forward shockSE,1, which interacts with the linex(t) =−(√
70/10)(t−tC) in G(xG, tG) generating a forward shock SG,1: ˙x(t) =σ 0, 1
2 1 +
√70 10
!!
, x(tG) =xG.
On Ω2, let F be the point whereRC,2 andSB,2 meet. From this point starts a forward shockSF,2, which interacts with the linex(t) =−(√
70/10)(t−tC) inH(xH, tH) generating a forward shock SH,2: ˙x(t) =σ 0, 1
2 1 +
√70 10
!!
, x(tH) =xH,
which reaches the junction at timetJ≈2.90, then Ω2 is empty. At timet=tJ the non-local effective receiving capacityQnl attains the valueg(1/4) = 1/4 and a further rarefaction appears on Ω1. Its values are given by
RJ,1(t, x) = 1 2
1− x t−tJ
, for −
√70
10 (t−tJ)< x≤0.
Such rarefaction interacts withSG,1at I(xI, tI) generating another shock SI,1: ˙x(t) =σ(0,RJ,1(t, x(t))), x(tI) =xI, which reaches the junction at timet=tK.
On Ω3, at time t=tJ starts the rarefactionRJ,3 given by RJ,3(t, x) = 1
2
1− x t−tJ
, for 0< x≤
√10
5 (t−tJ),
and finally at timet=tK ≈4.15 a further shock starts and interacts withRJ,3generating the shock SK,3: ˙x(t) =σ(RJ,3(t, x(t)),0), x(tK) = 0.
5. Finite volumes numerical scheme for the constrained problem
In this section we describe a finite volumes numerical scheme, which can be used to construct solutions for the Cauchy problem at a junction with capacity drop representation. Our scheme is developed starting from the scheme introduced in [1].
In [26] it is shown that the scheme captures the correct solution on a merge where the flux through the junction is not constrained. Then, we show that our implementation of local and non-local point constraints is correct by comparison with the explicit solution computed in Sections3.1and4.1.
After that, we turn our attention to the comparison betweenRHBCj ,Rlj andSjnl. We reproduce the numerical simulation made in [22], then we run a simulation starting with the same initial conditions but using the non- local constraint at the junction. In this part we can notice that the capacity drop representation based on non-local point constraint allows to capture a more realistic behavior as the congestion disappears in finite time. Additionally, for a given constraint function g, we discuss the relation between the qualitative behavior of the numerical solution and the choice of the weight functionw.
5.1. Numerical scheme with constraint at the junction
We fix a constant space step ∆x. For ` ∈ Z and h ∈ H, we set xh` = `∆x. We define the cell centers xh`+1
2
= (`+12)∆xfor`∈Zand consider the uniform spatial mesh on each Ωh [
`≤−1
[xi`, xi`+1), i∈I, [
`≥0
[x3`, x3`+1),
so that the position of the junctionx= 0 corresponds toxh0 for each road. Then we fix a constant time step ∆t satisfying the CFL condition
∆tmax
h∈HLh≤∆x 2 ,
whereLh is the Lipschitz constant offh. Fors∈Nwe define the time discretizationts=s∆t. At each timets, ρh,s`+1
2
represents an approximation of the mean value of the solution on the interval [xh`, xh`+1),`∈Z, along the hth road.
We initialize the scheme by discretizing the initial conditions
ρh,0`+1 2
= 1
∆x Z xh`+1
xh`
ρh,0(x) dx, for allh∈Hand for`≤ −1 ifh∈I,`≥0 ifh= 3.
For each s ∈ N, at all cell interfaces xh` with ` 6= 0, we consider a monotone, consistent numerical flux Fh(ρh,s`−1/2, ρh,s`+1/2) corresponding to the fluxfh. At the junctionxh0 we take on each road Ωhthe Godunov flux Ghcorresponding to the solution of the Riemann problem at the junction computed by the appropriate solver.
Then, the finite volumes scheme can be computed by a two-step procedure:
(i) find
( ˆρ1,ρˆ2,ρˇ3) such that fi( ˆρi) = Γi fori∈I, and f3( ˇρ3) = Γ1+ Γ2, (5.1) where Γ1 and Γ2 are defined in (2.5);
(ii) compute
ρh,s+1`+1 2
=ρh,s`+1 2
− ∆t
∆x
F`+1h,s − F`h,s
, (5.2)
where
F`h,s=
Fh
ρh,s`−1/2, ρh,s`+1/2
ifh∈I and`≤ −1 or h= 3 and`≥1, Gh
ρh,s
−12,ρˆh
ifh∈I and`= 0, Gh
ˇ ρ3, ρh,s1
2
ifh= 3 and `= 0,
(5.3)
and Fh
ρh,s`−1/2, ρh,s`+1/2
is a monotone, consistent numerical flux, i.e.for allh∈H – Fh is Lipschitz continuous from [0, ρmax]2 toR,
– Fh(a, a) =fh(a) for anya∈[0, ρmax],
– the map (a, b)∈[0, ρmax]27→Fh(a, b)∈R is non-decreasing with respect toa and non-increasing with respect tob.
In principle any monotone and consistent numerical flux might be used away from the junction, but we limit our attention to Godunov flux.
Notice that the choice ~ρof the implementation of the local or non-local point constraint happens when we compute the boundary data in (5.1). In particular, when we deal with the non-local point constraint, we need to approximate the weighted average of the densityζi,i∈Ias follows
Zis= ∆xX
`≤0
wi(x`+1
2)ρi,s`+1 2
, i∈I.
Moreover, when we implement the local point constraint, we can apply Proposition 3.1 in order to find the boundary data in (5.1) corresponding toRlj. However, in our simulations we implementRHBCj , indeed, as already observed, after few time iterations we observe the solution corresponding toRlj.
5.2. Validation of the numerical scheme
The implementation of the scheme described in [1] for a merge without capacity drop representation has been done in [26].
